Tagged Questions
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2
votes
0answers
66 views
+100
Generating random weak k-bounded reverse plane partitions
Fix a partition $\lambda$. A weak reverse plane partition of shape $\lambda$ is a filling $0\leq \pi_{ij}$ of $\lambda$ with $\pi_{ij}\leq \pi_{kl}$ whenever $i\leq k$ or $j\leq l$. Note that ...
6
votes
3answers
276 views
Representations of S_n induced from centralizers of elements
Does anyone have a reference for a good description of representations of $S_{n}$ obtained by inducing up from $C_{S_{n}}(\pi)$, for some element $\pi$ of $S_{n}$? (I'd prefer an efficient ...
2
votes
1answer
156 views
Running the Greene-Nijenhuis Algorithm Backwards
This question is crossposted from math.stackexchange.com, where it remains unanswered.
Let $Y$ be a Young tableau of shape $\lambda:=(\lambda_1,\ldots,\lambda_n)$, where ...
12
votes
2answers
337 views
calculating Littlewood-Richardson coefficients
It is known that if $\alpha,\beta,\gamma$ are three partitions then the Littlewood-Richardson coefficient $c_{\alpha \beta}^{\gamma}$ is positive when the triple ($\alpha,\beta,\gamma$
) occurs as ...
14
votes
2answers
763 views
Has Reifegerste's Theorem on RSK and Knuth relations received a slick proof by now?
For the notations I am using, I refer to the Appendix at the end of this post.
Here is what, for the sake of this post, I consider to be Reifegerste's theorem:
Theorem 1. Let $n\in\mathbb N$ and ...
6
votes
0answers
210 views
What are the homological properties of Young's lattice?
Young's lattice $Y$ is a graded poset and a distributive lattice whose elements are all the partitions of $n$ for $n \in \mathbb{N}$ with the poset relation coming from inclusion of Young diagrams. ...
1
vote
2answers
202 views
Flips on standard Young tableaux and descent sets
Consider $T$ to be a standard Young tableau of shape $\lambda$ (in English notation). The descent set of $T$, $Des(T)$, is defined to the set of all positive integers $i$ such that $i+1$ lies strictly ...
14
votes
1answer
405 views
Conjectural identities for Young symmetrizers and Young-Jucys-Murphy elements
The following questions I have found in my own notes from about 3 years ago. Unfortunately, I lost much of the context; I believe I made these conjectures reading Okounkov-Vershik, arXiv:0503040v3, ...
7
votes
1answer
253 views
Does this lattice have a name (and literature)?
The "lattice" in the title appears to be a lattice. At least it's a poset, which I define now.
Fix a partition $\lambda$ of $n$ and consider the set of all standard Young tableaux (each of ...
7
votes
1answer
348 views
On Applications of Murnaghan Nakayama Rule
This question is crossposted at math.stackexchange here and may be beyond the usual scope of the site. The question is located below. In short, I am looking for an accessible explanation of the ...
3
votes
1answer
118 views
Random RSK and Plancherel Measure
Let $(X_1,X_2,\ldots)$ be a sequence of i.i.d. random variables. It is known that if these random variables are distributed uniformly on the unit interval, then applying the RSK algorithm to this ...
4
votes
1answer
183 views
Expression of basis vectors of permutation modules in different bases.
This is a cross-post from math.se, because I did not get any answer there:
Write $[n]:=\{1,\ldots,n\}$. For a partition $\lambda\vdash n$, I will write $[\lambda]$ for the Specht module that ...
9
votes
2answers
241 views
How exactly does Schützenberger promotion relate to Striker-Williams promotion?
Schützenberger promotion, studied (for example) in Richard Stanley, Promotion and Evacuation, 2009, is a permutation of the set of all linear extensions of a finite poset. Since one can identify the ...
10
votes
1answer
397 views
Generating Random Young Tableaux: A peculiar probability identity
In the paper by Greene, Nijenhuis and Wilf, an algorithm is proposed for generating uniformly random Young tableaux of shape $\lambda$. The algorithm is to uniformly randomly pick a starting cell, and ...
4
votes
1answer
161 views
Representations of Sym(n) and SL_d
Irreducible representations of the symmetric group Sym$(n)$, and degree-$n$ algebraic representations of SL$_d(\mathbb C)$ for $d\ge n$, can both be classified by Young diagrams with $n$ boxes.
...
8
votes
3answers
543 views
bijection between number of partitions of 2n satisfying certain conditions with number of partitions of n
Suppose $\lambda = (\lambda_1,\lambda_2,.....,\lambda_k)$ is a partition of $2n$ where $n \in \mathbb N$ satisfying the following conditions:
(1) $\lambda_{k} = 1$.
(2) $\lambda_{i} - \lambda_{i+1} ...
5
votes
0answers
177 views
Explicit description of isomorphism when decomposing into irreps
I had a question which is slightly more general than this one on mathoverflow: I am looking for an explicit description of the isomorphism
$\mathbb S_\nu(V\otimes W) \cong \bigoplus C_{\lambda\mu\nu} ...
6
votes
0answers
132 views
An inequality for the ratio of standard Young tableau with {1,2,…,k} in the first row
For a partition $\lambda \vdash n$, define $\dim \lambda$ to be the number of standard Young tableaux of shape $\lambda$, and $\dim \lambda/\(k)$ as the number of standard Young tableaux with ...
6
votes
2answers
288 views
Viennot-type geometric description for dual RSK correspondence?
Is a geometric construction of the dual RSK correspondence along the lines of Viennot's "light and shadows construction" written up somewhere? This is a bijective correspondence between 0-1 matrices ...
14
votes
1answer
466 views
Bruhat order and the Robinson-Schensted correspondence
The Robinson-Schensted correspondence is a bijection between elements of the symmetric group $S_n$ and pairs of standard tableaux of the same shape. The symmetric group is partially ordered by the ...
12
votes
1answer
320 views
Categorifying the equality of product and coproduct of symmetric functions
Littlewood-Richardson coefficients are both multiplicities of $GL_n$ tensor products, and of restrictions of $GL_{m+n}$ representations to $GL_m \times GL_n$. I want to turn this equality of numbers ...
4
votes
3answers
463 views
RS to RSK correspondence
The RS correspondence is a correspondence which associates to each permutation a pair of standard Young tableaux of the same shape.
The RSK correspondence associates to each integer matrix (with ...
2
votes
1answer
172 views
Box-dual of a partition - what is it called?
Fix natural numbers $n,m\in\mathbb{N}$. Given a partition $\lambda\vdash d$ with at most $n$ rows (and at most $m$ columns), we can define a partition ...
10
votes
1answer
431 views
Analogues of the Knuth and Forgotten equivalences on permutations: have they been studied?
Consider a totally ordered alphabet $A$ of $n$ letters. Let $W$ be the set of all words over $A$ which have no two letters equal. Then, for example, we can define the Knuth equivalence on $W$ as the ...
6
votes
0answers
242 views
The number of rows in a tableau generated by the RSK algorithm.
It is well known that the number of rows in the semistandard Young tableaux correspondent to a two-line array via RSK is equal to the length of the longest (strictly) decreasing subsequence in the ...
4
votes
0answers
148 views
Bound on coefficients in Young tableuax
The following may be known, but I didn't find anything in the literature.
Background:
The irreducible representations of $S_n$ correspond to shapes of Young tableaux with $n$ elements. Let $\lambda$ ...
2
votes
1answer
160 views
Dimension of spaces of invariants/tableaux functions
The Hook lenght formula gives the number of standard Young tableaux on a given diagram.
A variant gives the number of semistandard tableuax.
Does there exist a formula for counting "weighted ...
6
votes
1answer
450 views
Interpreting the Garnir relations in terms of the Yang-Baxter equation
One possible construction of the Specht modules goes as follows.
Given a partition $\lambda$ of $n$, we can write down Young's seminormal form for the representation of $S_n$ corresponding to ...
11
votes
2answers
836 views
Sym(V ⊕ ∧² V) isomorphic to direct sum of all Schur functors of V
Let $V$ be a finite-dimensional $K$-vector space. Then, the symmetric power $\mathrm{Sym}\left(V\oplus \wedge^2 V\right)$ is isomorphic to the direct sum of all Schur functors applied to $V$ (each one ...
3
votes
1answer
263 views
Is there an account of the algebra of highest weight tensors?
The context for this question is Schur-Weyl duality. Let $V$ be a vector space. For $r>0$ consider $\otimes^rV$. This has commuting actions of the symmetric group $S(r)$ and $G=GL(V)$.
My question ...
4
votes
2answers
761 views
A sum involving irreducible characters of the symmetric group
Recently, during the research, I came across a sum, denoted by $H(n,L)$, involving irreducible characters of the symmetric group,
\begin{equation}
H(n,L)\colon=\sum_{Y_{i,j,w}} ...
4
votes
1answer
590 views
some confusion about the explicit construction of irreducible representations of $S_n$
In this book chapter, the irreducible representations of the symmetric group $S_n$ is given in terms of polytabloids of a Ferrer's diagram $\lambda$, defined as
$e_t = \sum_{\pi \in C_t} ...
10
votes
4answers
1k views
Making the branching rule for the symmetric group concrete
This question concerns the characteristic $0$ representation theory of the symmetric group $S_n$. I'm a topologist, not a representation theorist, so I apologize if I state it in an odd way.
First, ...
6
votes
2answers
654 views
What is the most general “two in one row for A & in one column for B” theorem?
Let $A$ and $B$ be two Young tableaux, i. e. Young diagrams filled with the numbers $1$, $2$, ..., $n$ for some $n$ (not necessarily the same $n$). (They need not be semistandard.)
(a) (Etingof's ...
12
votes
3answers
507 views
What bijection on permutations corresponds under RS to transpose?
The Robinson-Schensted correspondence is a bijection between elements of the symmetric group and ordered pairs of standard tableaux of the same shape.
Some simple operations on tableaux correspond to ...
5
votes
1answer
216 views
Asymptotics of symmetry types of tensors
Introduction
Let's fix $m\in \mathbb N$. For each n, the unitary group $\mathbf U(m)$ is represented in the space of tensors of rank $n$ over $\mathbb C^m$
$$V_{n,m}=\bigotimes_{k=1}^n \mathbb C^m$$ ...
11
votes
3answers
1k views
Equivalence relations on permutations and pattern avoidance
I'm working on the interaction between equivalence relations on permutations and pattern avoidance. I've only considered Knuth equivalence and cyclic shifts until now and I'm looking for other ...
19
votes
3answers
699 views
Statistics of irreps of S_n that can be read off the Young diagram, and consequences of Kerov-Vershik
Alexei Oblomkov recently told me about the beautiful theorem of Kerov and Vershik, which says that "almost all Young diagrams look the same." More precisely: take a random irreducible representation ...
7
votes
3answers
600 views
Symmetric Powers, Tableau and Wreath Products
Let V and W be irreducible representations of $S_n$ and $S_m$ over a field of characteristic 0. Then the Littlewood-Richardson coefficients allow us to compute the isomorphism type of the induced ...
14
votes
3answers
852 views
Young's lattice and the Weyl algebra
Let L be the lattice of Young diagrams ordered by inclusion and let Ln denote the nth rank, i.e. the Young diagrams of size n. Say that lambda > mu if lambda covers mu, i.e. mu can be obtained ...
